L11a463

L11a462

L11a464

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a463 at Knotilus!

Link Presentations

[edit Notes on L11a463's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X8,14,9,13 X22,8,13,7 X20,15,21,16 X16,6,17,5 X18,12,19,11 X12,18,5,17 X10,20,11,19 X2,9,3,10 X4,22,1,21
Gauss code	{1, -10, 2, -11}, {6, -1, 4, -3, 10, -9, 7, -8}, {3, -2, 5, -6, 8, -7, 9, -5, 11, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^3 w^2-2 u v^3 w+u v^2 w^3-3 u v^2 w^2+4 u v^2 w-2 u v^2-2 u v w^3+4 u v w^2-3 u v w+u v+u w^3-2 u w^2+u w-v^3 w^2+2 v^3 w-v^3-v^2 w^3+3 v^2 w^2-4 v^2 w+2 v^2+2 v w^3-4 v w^2+3 v w-v+2 w^2-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$q^9-3 q^8+7 q^7-9 q^6+15 q^5-17 q^4+17 q^3-15 q^2+12 q-7+4 q^{-1} - q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2+ a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db)
Kauffman polynomial	$z^6 a^{-10} -3 z^4 a^{-10} +z^2 a^{-10} +3 z^7 a^{-9} -8 z^5 a^{-9} +2 z^3 a^{-9} +6 z^8 a^{-8} -22 z^6 a^{-8} +28 z^4 a^{-8} -20 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +5 z^9 a^{-7} -13 z^7 a^{-7} +5 z^5 a^{-7} +11 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +4 z^8 a^{-6} -30 z^6 a^{-6} +49 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +10 z^9 a^{-5} -30 z^7 a^{-5} +32 z^5 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +4 z^8 a^{-4} -18 z^6 a^{-4} +24 z^4 a^{-4} -12 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +5 z^9 a^{-3} -8 z^7 a^{-3} +9 z^5 a^{-3} -7 z^3 a^{-3} +6 z^8 a^{-2} -7 z^6 a^{-2} -z^4 a^{-2} +6 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+z^3 a^{-1} +4 z^6-7 z^4+2 z^2+1$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3 1 -2 15                   4     4 13                 5 3     -2 11               10 4       6 9             8 6         -2 7           9 9           0 5         6 8             2 3       6 9               -3 1     3 8                 5 -1   1 4                   -3 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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